Time Hierarchies with One Bit of Advice Konstantin Pervyshev Steklov Mathematics Institute, St. Petersburg, Russia joint work with Dieter van Melkebeek

Time Hierarchy An open question for probabilistic algorithms: is there a time hierarchy ? O(n 2 ) O(n 3 ) O(n) (we still can’t disprove this)

Our result Previous results –[Barak 02] uses a modified notion of algorithm (algorithms with small advice) –Under the modified notion of algorithm, a time hierarchy for the probabilistic algorithms exists Our result –Under the same modified notion of algorithm, a time hierarchy exists for any class of algorithms various classes of probabilistic algorithms Arthur-Merlin and Merlin-Arthur games, other kinds of IP NP ∩ co-NP

Outline Why standard techniques don’t work Where advice helps How to prove the generic time hierarchy

Diagonalization To separate deterministic time n a+e from time n a, consider a machine M such that –M(k) := not N k (k), where n a steps of N k are simulated M runs in time n a+ M recognizes some languages L L can’t be recognized in time n a (!) deterministic algorithms are recursively enumerable

Probabilistic Algortithms and Diagonalization Probabilistic Algorithms –Probabilistic Turing machines that satisfy some condition on the error probability Two-sided error (BPP) –Pr[M(x) = 1] > 2/3 or Pr[M(x) = 0] > 2/3 –“machine M is good on input x” –“machine M is good at length n” Need to enumerate only good machines M(k) := not N k (k) Pr[N k (k) = 1] = 1/2 => Pr[M(k) = 1] = 1/2 It’s not possible

More Failures Various classes of probabilistic algorithms –bounded probability of error BPP – two-sided error RR – one-sided error ZPP – zero-sided error NP ∩ co-NP –two machines solve the same language Generally speaking, semantic classes Diagonalization fails –N k (k) is bad => M(k) is bad To overcome this, M needs advice on whether N k is good

Algorithms with Advice Turing machine M on input x of length n is provided with –some advice a(n) of length l(n) Advice is the same for every input of length n Depending on the advice provided, –M may recognize several languages –M may satisfy the promise or not Advice of length 1 bit helps with time hierarchies

Time Hierarchies with Advice A time hierarchy exists for probabilistic algorithms with advice of length –O(log log n) bits – [Barak 02] –1 bit – [Fortnow, Santhanam 04] Time hierarchy for any class of algorithms with advice of length –O(log n * log log n) bits – [Fortnow, Santhanam, Trevisan 05] –1 bit – our result

Generic Time Hierarchy –To separate n a+e from n a, it’s sufficient to prove that for any 1 ≤ a, there exists a language L solvable in probabilistic polynomial time with 1 bit of advice – machine M with advice a(n) not solvable in probabilistic time n a with 1 bit of advice –any machine N k with any advice b(n)

A Failed Approach Construct M with advice a(n) so that –for some inputs x (0) and x (1) of the same length n M (x (0),a(n)) := not N k (x (0),0) M (x (1),a(n)) := not N k (x (1),1) Both N k (x,0) and N k (x,1) may be bad => M needs 2 bits of advice in order to diagonalize safely

Another Failed Approach M can safely simulate N k via deterministic simulation –needs exponentially more time To get exponentially more time, we use delayed diagonalization

A Step of Delayed Diagonalization x (0) x (1) z (1) y (0) Advice on whether N/0 is good on x’s M(z (1) ) = N(x (1),1) M(y (0) ) = “no” Advice on whether N/1 is good on x’s N/0 is bad N/1 is good

Tree-Like Delayed Diagonalization x (00-11) z (10,11) y (00,01) v (01) w (11) M(z (01) ) = N(x (01),1) M(z (11) ) = N(x (11),1) M(v (01) ) = N(z (01),0) M(y (00) ) = “no” M(y (01) ) = “no” M(w (11) ) = “no” N/0 is bad N/1 is good N/0 is good N/1 is bad

Towards a Contradiction –Assume for some advice b(n), N is good and solves the same language as M –Then N(v (s),b(|v|)) = M(v (s),a(|v|)) = N(z (s),b(|z|)) = M(z (s),a(|z|)) = N(x (s),b(|x|)) = M(x (s),a(|x|)) –Therefore, N(v (s),b(|v|)) = M(x (s),a(|x|)) for some s –So let M(x (s),a(|x|)) := not N(v (s),b(|v|)) this can be done deterministically –thus a contradiction x (s) z (s) v (s) |x| ~ 2 |v| a

Choice of the Input Lengths We need –parent’s length is polynomial in children’s length so that M runs in poly-time –for any leaf v, roots length is greater than 2 |v| a so that M can deterministically simulate N at leaves It’s possible to satisfy these conditions QED x (s) z (s) v (s)

Summary A time hierarchy exists for virtually any kind of algorithms with one bit of advice The probabilistic time hierarchy with advice is a property of algorithms with advice

Thank you! Dieter van Melkebeek, Konstantin Pervyshev “A Generic Time Hierarchy for Semantic Models with One Bit of Advice” (CCC’06)

Generic Time Hierarchy –Theorem for any 1 ≤ a < b, there exists a language L solvable in probabilistic time n b with 1 bit of advice not solvable in probabilistic time n a with 1 bit of advice –Only basic properties of algorithms are needed –Approach Construct probabilistic M with 1-bit advice a(n) that –works in time n b –is good Prove that for any probabilistic N with any 1-bit advice b(n) that –works in time n a –is good There exists x such that M(x,a(|x|)) ≠ N(x,b(|x|))

Non-Uniform World Previous results –[Barak02, FS04, FST05] a time hierarchy exists for 1 bit non-uniform probabilistic algorithms with two- and one- sided error Our result –a time hierarchy exists for any class of 1 bit non-uniform algorithms various classes of probabilistic algorithms Arthur-Merlin and Merlin-Arthur games, other kinds of IP NP ∩ co-NP

Recall Non-deterministic Time Hierarchy Non-deterministic time –M can copy (simulate) N –M can’t negate N Our case –M can copy N(x,b(|x|)) –We have no idea of how to negate N trivially Delayed diagonalization x (0) x (1) z (1) y (0)